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			<h1 id="firstHeading" class="firstHeading">Whittaker–Shannon interpolation formula</h1>
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<td class="mbox-text" style="">This article <b>does not <a href="http://en.wikipedia.org/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="http://en.wikipedia.org/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">references or sources</a></b>.<br>
<small>Please help <a href="http://en.wikipedia.org/w/index.php?title=Whittaker%E2%80%93Shannon_interpolation_formula&amp;action=edit" class="external text" rel="nofollow">improve this article</a> by adding citations to <a href="http://en.wikipedia.org/wiki/Wikipedia:Identifying_reliable_sources" title="Wikipedia:Identifying reliable sources">reliable sources</a>. Unsourced material may be <a href="http://en.wikipedia.org/wiki/Template:Citation_needed" title="Template:Citation needed">challenged</a> and <a href="http://en.wikipedia.org/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>. <i>(December 2009)</i></small></td>
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<p>The <b>Whittaker–Shannon interpolation formula</b> is a method to reconstruct a <a href="http://en.wikipedia.org/wiki/Continuous-time" title="Continuous-time" class="mw-redirect">continuous-time</a> <a href="http://en.wikipedia.org/wiki/Bandlimited" title="Bandlimited" class="mw-redirect">bandlimited</a> signal from a set of equally spaced samples.</p>
<table id="toc" class="toc">
<tbody><tr>
<td>
<div id="toctitle">
<h2>Contents</h2>
 <span class="toctoggle">[<a href="#" class="internal" id="togglelink">hide</a>]</span></div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="#Definition"><span class="tocnumber">1</span> <span class="toctext">Definition</span></a></li>
<li class="toclevel-1 tocsection-2"><a href="#Validity_condition"><span class="tocnumber">2</span> <span class="toctext">Validity condition</span></a></li>
<li class="toclevel-1 tocsection-3"><a href="#Interpolation_as_convolution_sum"><span class="tocnumber">3</span> <span class="toctext">Interpolation as convolution sum</span></a></li>
<li class="toclevel-1 tocsection-4"><a href="#Convergence"><span class="tocnumber">4</span> <span class="toctext">Convergence</span></a></li>
<li class="toclevel-1 tocsection-5"><a href="#Stationary_random_processes"><span class="tocnumber">5</span> <span class="toctext">Stationary random processes</span></a></li>
<li class="toclevel-1 tocsection-6"><a href="#See_also"><span class="tocnumber">6</span> <span class="toctext">See also</span></a></li>
</ul>
</td>
</tr>
</tbody></table>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Whittaker%E2%80%93Shannon_interpolation_formula&amp;action=edit&amp;section=1" title="Edit section: Definition">edit</a>]</span> <span class="mw-headline" id="Definition">Definition</span></h2>
<p>The <b>interpolation formula</b>, as it is commonly called, dates back to works of <a href="http://en.wikipedia.org/wiki/E._Borel" title="E. Borel" class="mw-redirect">E. Borel</a> in 1898, and <a href="http://en.wikipedia.org/wiki/E._T._Whittaker" title="E. T. Whittaker">E. T. Whittaker</a> in 1915, and was cited from works of <a href="http://en.wikipedia.org/wiki/J._M._Whittaker" title="J. M. Whittaker" class="mw-redirect">J. M. Whittaker</a> in 1935 in the formulation of the <a href="http://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem" title="Nyquist–Shannon sampling theorem">Nyquist–Shannon sampling theorem</a> by <a href="http://en.wikipedia.org/wiki/Claude_Shannon" title="Claude Shannon">Claude Shannon</a> in 1949. It is also commonly called <b>Shannon's interpolation formula</b> and <b>Whittaker's interpolation formula</b>. E. T. Whittaker, who published it in 1915, called it the <b>Cardinal series</b>.</p>
<p>The <a href="http://en.wikipedia.org/wiki/Sampling_theorem" title="Sampling theorem" class="mw-redirect">sampling theorem</a> states that, under certain <i>limiting conditions</i>, a function <i>x</i>(<i>t</i>) can be recovered exactly from its samples, &nbsp; <i>x</i>[<i>n</i>] = <i>x</i>(<i>nT</i>), by the Whittaker–Shannon interpolation formula<b>:</b></p>
<dl>
<dd><img class="tex" alt="x(t) = \sum_{n=-\infty}^{\infty} x[n] \cdot {\rm sinc}\left(\frac{t - nT}{T}\right)\," src="wikipedia-Whittaker%E2%80%93Shannon_interpolation_formula_pliki/946207f70f194bb9c56133577ffc0144.png"></dd>
</dl>
<p>where <i>T</i> = 1/<i>f</i><sub><i>s</i></sub> is the <i>sampling interval</i>, <i>f</i><sub><i>s</i></sub> is the <i><a href="http://en.wikipedia.org/wiki/Sampling_rate" title="Sampling rate">sampling rate</a></i>, and sinc(<i>x</i>) is the normalized <a href="http://en.wikipedia.org/wiki/Sinc_function" title="Sinc function">sinc function</a>.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Whittaker%E2%80%93Shannon_interpolation_formula&amp;action=edit&amp;section=2" title="Edit section: Validity condition">edit</a>]</span> <span class="mw-headline" id="Validity_condition">Validity condition</span></h2>
<div class="thumb tright">
<div class="thumbinner" style="width:242px;"><a href="http://en.wikipedia.org/wiki/File:Bandlimited.svg" class="image"><img alt="" src="wikipedia-Whittaker%E2%80%93Shannon_interpolation_formula_pliki/240px-Bandlimited.png" class="thumbimage" height="122" width="240"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Bandlimited.svg" class="internal" title="Enlarge"><img src="wikipedia-Whittaker%E2%80%93Shannon_interpolation_formula_pliki/magnify-clip.png" alt="" height="11" width="15"></a></div>
Spectrum of a <b>bandlimited signal</b> as a function of frequency. The two-sided bandwidth <i>R</i><sub><i>N</i></sub> = 2<i>B</i> is known as the Nyquist rate for the signal.</div>
</div>
</div>
<p>If the function <i>x</i>(<i>t</i>) is bandlimited, and sampled at a 
high enough rate, the interpolation formula is guaranteed to reconstruct
 it exactly. Formally, if there exists some <i>B</i> ≥ 0 such that</p>
<ol>
<li>the function <i>x</i>(<i>t</i>) is bandlimited to bandwidth <i>B</i>; that is, it has a <a href="http://en.wikipedia.org/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> <img class="tex" alt="\scriptstyle \mathcal{F} \{x(t) \} = X(f) = 0 \ " src="wikipedia-Whittaker%E2%80%93Shannon_interpolation_formula_pliki/f763eabe48d1654050df68d42a72a238.png"> for |<i>f</i>| &gt; <i>B</i>; and</li>
<li>the <a href="http://en.wikipedia.org/wiki/Sampling_rate" title="Sampling rate">sampling rate</a>, <i>f</i><sub><i>s</i></sub>, exceeds the <a href="http://en.wikipedia.org/wiki/Nyquist_rate" title="Nyquist rate">Nyquist rate</a>, twice the bandwidth: <i>f</i><sub><i>s</i></sub> &gt; 2<i>B</i>. Equivalently<b>:</b></li>
</ol>
<dl>
<dd>
<dl>
<dd><img class="tex" alt="T &lt; \frac{1}{2B}; " src="wikipedia-Whittaker%E2%80%93Shannon_interpolation_formula_pliki/6ff35c1a73fdbc55b679b9bdc3db76cf.png"></dd>
</dl>
</dd>
</dl>
<p>then the interpolation formula will exactly reconstruct the original <i>x</i>(<i>t</i>) from its samples. Otherwise, aliasing may occur; that is, frequencies at or above <i>f</i><sub><i>s</i></sub>/2 may be erroneously reconstructed. <i>See <a href="http://en.wikipedia.org/wiki/Aliasing" title="Aliasing">Aliasing</a> for further discussion on this point.</i></p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Whittaker%E2%80%93Shannon_interpolation_formula&amp;action=edit&amp;section=3" title="Edit section: Interpolation as convolution sum">edit</a>]</span> <span class="mw-headline" id="Interpolation_as_convolution_sum">Interpolation as convolution sum</span></h2>
<p>The interpolation formula is derived in the <a href="http://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem" title="Nyquist–Shannon sampling theorem">Nyquist–Shannon sampling theorem</a> article, which points out that it can also be expressed as the <a href="http://en.wikipedia.org/wiki/Convolution" title="Convolution">convolution</a> of an <a href="http://en.wikipedia.org/wiki/Dirac_comb" title="Dirac comb">infinite impulse train</a> with a sinc function<b>:</b></p>
<dl>
<dd><img class="tex" alt=" x(t) = \left( \sum_{n=-\infty}^{\infty} x[n]\cdot \delta \left( t - nT \right) \right) * 
{\rm sinc}\left(\frac{t}{T}\right). " src="wikipedia-Whittaker%E2%80%93Shannon_interpolation_formula_pliki/3937f3a9594cae525350ef02090f275b.png"></dd>
</dl>
<p>This is equivalent to filtering the impulse train with an ideal (<i>brick-wall</i>) <a href="http://en.wikipedia.org/wiki/Low-pass_filter" title="Low-pass filter">low-pass filter</a>.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Whittaker%E2%80%93Shannon_interpolation_formula&amp;action=edit&amp;section=4" title="Edit section: Convergence">edit</a>]</span> <span class="mw-headline" id="Convergence">Convergence</span></h2>
<p>The interpolation formula always converges <a href="http://en.wikipedia.org/wiki/Absolut_convergence" title="Absolut convergence" class="mw-redirect">absolutely</a> and <a href="http://en.wikipedia.org/wiki/Uniform_convergence" title="Uniform convergence">locally uniform</a> as long as</p>
<dl>
<dd><img class="tex" alt="\sum_{n\in\Z,\,n\ne 0}\left|\frac{x[n]}n\right|&lt;\infty." src="wikipedia-Whittaker%E2%80%93Shannon_interpolation_formula_pliki/dddf39c1f4ff3924322a4254eefdb237.png"></dd>
</dl>
<p>By the <a href="http://en.wikipedia.org/wiki/H%C3%B6lder_inequality" title="Hölder inequality" class="mw-redirect">Hölder inequality</a> this is satisfied if the sequence <img class="tex" alt="\scriptstyle (x[n])_{n\in\Z}" src="wikipedia-Whittaker%E2%80%93Shannon_interpolation_formula_pliki/75410a026c34daba17a2b32d2b5bbecc.png"> belongs to any of the <img class="tex" alt="\scriptstyle\ell^p(\Z,\mathbb C)" src="wikipedia-Whittaker%E2%80%93Shannon_interpolation_formula_pliki/b2ac6bc03ebb86bf528a2bb2e2eaac80.png"> <a href="http://en.wikipedia.org/wiki/Lp_space" title="Lp space">spaces</a> with 1&nbsp;&lt;&nbsp;<i>p</i>&nbsp;&lt;&nbsp;∞, that is</p>
<dl>
<dd><img class="tex" alt="\sum_{n\in\Z}\left|x[n]\right|^p&lt;\infty." src="wikipedia-Whittaker%E2%80%93Shannon_interpolation_formula_pliki/04b6d74a1819d7c7506a2f30c2ecd0a4.png"></dd>
</dl>
<p>This condition is sufficient, but not necessary. For example, the sum
 will generally converge if the sample sequence comes from sampling 
almost any <a href="http://en.wikipedia.org/wiki/Stationary_process" title="Stationary process">stationary process</a>, in which case the sample sequence is not square summable, and is not in any <img class="tex" alt="\scriptstyle\ell^p(\Z,\mathbb C)" src="wikipedia-Whittaker%E2%80%93Shannon_interpolation_formula_pliki/b2ac6bc03ebb86bf528a2bb2e2eaac80.png"> space.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Whittaker%E2%80%93Shannon_interpolation_formula&amp;action=edit&amp;section=5" title="Edit section: Stationary random processes">edit</a>]</span> <span class="mw-headline" id="Stationary_random_processes">Stationary random processes</span></h2>
<p>If <i>x</i>[<i>n</i>] is an infinite sequence of samples of a sample function of a wide-sense <a href="http://en.wikipedia.org/wiki/Stationary_process" title="Stationary process">stationary process</a>, then it is not a member of any <img class="tex" alt="\scriptstyle\ell^p" src="wikipedia-Whittaker%E2%80%93Shannon_interpolation_formula_pliki/8a384ee57e05dab645c6ba8e475b190f.png"> or <a href="http://en.wikipedia.org/wiki/Lp_space" title="Lp space">L<sup>p</sup> space</a>, with probability 1; that is, the infinite sum of samples raised to a power <i>p</i>
 does not have a finite expected value. Nevertheless, the interpolation 
formula converges with probability 1. Convergence can readily be shown 
by computing the variances of truncated terms of the summation, and 
showing that the variance can be made arbitrarily small by choosing a 
sufficient number of terms. If the process mean is nonzero, then pairs 
of terms need to be considered to also show that the expected value of 
the truncated terms converges to zero.</p>
<p>Since a random process does not have a Fourier transform, the 
condition under which the sum converges to the original function must 
also be different. A stationary random process does have an <a href="http://en.wikipedia.org/wiki/Autocorrelation_function" title="Autocorrelation function" class="mw-redirect">autocorrelation function</a> and hence a <a href="http://en.wikipedia.org/wiki/Spectral_density" title="Spectral density">spectral density</a> according to the <a href="http://en.wikipedia.org/wiki/Wiener%E2%80%93Khinchin_theorem" title="Wiener–Khinchin theorem">Wiener–Khinchin theorem</a>.
 A suitable condition for convergence to a sample function from the 
process is that the spectral density of the process be zero at all 
frequencies equal to and above half the sample rate.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Whittaker%E2%80%93Shannon_interpolation_formula&amp;action=edit&amp;section=6" title="Edit section: See also">edit</a>]</span> <span class="mw-headline" id="See_also">See also</span></h2>
<ul>
<li><a href="http://en.wikipedia.org/wiki/Aliasing" title="Aliasing">Aliasing</a>, <a href="http://en.wikipedia.org/wiki/Anti-aliasing_filter" title="Anti-aliasing filter">Anti-aliasing filter</a>, <a href="http://en.wikipedia.org/wiki/Spatial_anti-aliasing" title="Spatial anti-aliasing">Spatial anti-aliasing</a></li>
<li><a href="http://en.wikipedia.org/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a></li>
<li><a href="http://en.wikipedia.org/wiki/Rectangular_function" title="Rectangular function">Rectangular function</a></li>
<li><a href="http://en.wikipedia.org/wiki/Sampling_%28signal_processing%29" title="Sampling (signal processing)">Sampling (signal processing)</a></li>
<li><a href="http://en.wikipedia.org/wiki/Signal_%28electronics%29" title="Signal (electronics)">Signal (electronics)</a></li>
<li><a href="http://en.wikipedia.org/wiki/Sinc_function" title="Sinc function">Sinc function</a>, <a href="http://en.wikipedia.org/wiki/Sinc_filter" title="Sinc filter">Sinc filter</a></li>
</ul>


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